Conservation laws and variational structure of damped nonlinear wave equations

TL;DR

This paper identifies all low-order conservation laws for a class of damped nonlinear wave equations in one dimension, revealing new variational symmetries and extending the understanding of their physical and mathematical structure.

Contribution

It derives all low-order conservation laws for damped nonlinear wave equations and uncovers novel variational symmetries not present in undamped cases.

Findings

Conservation laws include generalized momentum, energy, and light-cone energies.

Conformal momentum and dilational energy are unique to damped equations.

New variational symmetries differ from those in undamped wave equations.

Abstract

All low-order conservation laws are found for a general class of nonlinear wave equations in one dimension with linear damping which is allowed to be time-dependent. Such equations arise in numerous physical applications and have attracted much attention in analysis. The conservation laws describe generalized momentum and boost momentum, conformal momentum, generalized energy, dilational energy, and light-cone energies. Both the conformal momentum and dilational energy have no counterparts for nonlinear undamped wave equations in one dimension. All of the conservation laws are obtainable through Noether's theorem, which is applicable because the damping term can be transformed into a time-dependent self-interaction term by a change of dependent variable. For several of the conservation laws, the corresponding variational symmetries have a novel form which is different than any of the well known variation symmetries admitted by nonlinear undamped wave equations in one dimension.